College Algebra, Chapter 1, 1.5, Section 1.5, Problem 74

Solve the equation $\sqrt{x} + a \sqrt[3]{x} + b \sqrt[6]{x} + ab = 0$. Suppose that $a$ and $b$ are positive real number.


$
\begin{equation}
\begin{aligned}

\sqrt{x} + a \sqrt[3]{x} + b \sqrt[6]{x} + ab =& 0
&& \text{Given}
\\
\\
(\sqrt{x} + a \sqrt[3]{x}) +(b \sqrt[6]{x} + ab) =& 0
&& \text{Group terms}
\\
\\
\sqrt[3]{x} (\sqrt[6]{x} + a) + b (\sqrt[6]{x} + a) =& 0
&& \text{Factor out } \sqrt[3]{x} \text{ and } b
\\
\\
(\sqrt[3]{x} + b)(\sqrt[6]{x} + a) =& 0
&& \text{Factor out } \sqrt[3]{x} + b
\\
\\
\sqrt[3]{x} + b =& 0 \text{ and } \sqrt[6]{x} + a = 0
&& \text{Zero Product Property}
\\\
\\
x =& (-b)^3 \text{ and } x = (-a)^6
&& \text{Solve for } x
\\
\\
x =& -b^3 \text{ and } x = a^6
&&


\end{aligned}
\end{equation}
$